Stochastic Stability

Google is a well–known search engine that employs the PageRank algorithm [9] to rank the relative importance of webpages based on the link structure of the World–Wide Web. Intutively, rankings are based on a “random–surfer” model. Despite its tremendous success, Langville and Meyer [8] have pointed out some of its drawbacks. As the web continues to grow the computation of page ranks becomes more and more expensive, leading to less frequent recomputation (currently, on the order of weeks), which in turn leads to “stale” values being used in Web queries. In order to insure convergence, the algorithm introduces random perturbations, which corrupts the information inherent in the Web’s link–structure. Anecdotally, it is also well–known that individuals may collude to increase their ranking by creating additional links to one another [1]. In addition, it tends to create multiple references to what is ultimately the same document because it gives excessive weight to “navigational” pages, such as the results of queries to other web search–engines, relative to the actual pages referenced. Kamvar, et. al. [4] have suggested exploiting the hierarchical structure of the web to make the computation of page ranks more efficient. However, since they are ultimately interested in computing the same PageRank values, their approach does not solve any of the other problems. In this paper, we will analyze the PageRank algorithm to explain exactly how and why PageRank values may be manipulated. We will show how the “random–surfer” interpretation is related to a voting model. In particular, we will propose an alternative model of “weighted–voters” and suggest why this may be superior. We will also provide a theoretical justification for Kamvar, et. al.’s work as a “hierarchical voting system”. By choosing appropriate “personalization” vectors and avoiding their final PageRank iteration, we can address the remaining deficiencies in the Brin and Page ranking system. This analysis will depend on a number of new techical results regarding (perturbed) Markov processes [2, 11]. Specifically, we will describe a quotient construction which may be used both to eliminate transient states or to collape any closed class down to a proper subset of itself. We will show that this construction is natural, in the sense of category theory, which will make it easy to compute in a number of important